<p>In this paper, multi-dimensional new positivity-preserving (NPP) methods with a bigger sufficient CFL number of 1/6 are proposed and rigorously proven for the high-order finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) schemes on structured meshes. The MR-WENO spatial reconstruction procedures using the three, four, and five spatial stencils are implemented to obtain the quartic, sextic, and octic polynomials, respectively, which can obtain uniformly high-order accuracies in smooth regions and keep essentially non-oscillatory properties near strong discontinuities. Then, the new cell averages vectors are redefined to construct a sequence of polynomials vectors with different degrees. Since a novel methodology is proposed to obtain the minimum values of arbitrarily high degree polynomials vectors over the cell, the positivity of their density and pressure is checked. If the negativity occurs, a compression limiter is employed to enable the positive density and pressure for the polynomials vectors over the whole cell and the positive density and pressure for lower degree polynomials vectors at the midpoint of the cell. The NPP methods provide a new way of rigorously proving a sufficient CFL number of 1/6 for the fifth-order, seventh-order, and ninth-order MR-WENO schemes, which is bigger than 1/12, 1/20, and 1/30 for the same order WENO schemes when applying the classical positivity-preserving (PP) methods (Zhang and Shu in J. Comput. Phys. 229: 8918-8934, 2010). The numerical experiments show that the high-order NPP MR-WENO schemes exhibit increased computational efficiency by saving about 10–70% CPU time than before.</p>